OPEN Different population dynamics in the supplementary ...stat.columbia.edu/~cunningham/pdf/LaraNCOMMS2018.pdf · SMA and motor cortex both shared a large signal previously shown

ARTICLE

Different population dynamics in thesupplementary motor area and motorcortex during reachingA.H. Lara1, J.P. Cunningham 2,3,4,5 & M.M. Churchland1,3,4,6

Neural populations perform computations through their collective activity. Different com-

putations likely require different population-level dynamics. We leverage this assumption

to examine neural responses recorded from the supplementary motor area (SMA) and

motor cortex. During visually guided reaching, the respective roles of these areas remain

unclear; neurons in both areas exhibit preparation-related activity and complex patterns of

movement-related activity. To explore population dynamics, we employ a novel “hypothesis-

guided” dimensionality reduction approach. This approach reveals commonalities but also

stark differences: linear population dynamics, dominated by rotations, are prominent in motor

cortex but largely absent in SMA. In motor cortex, the observed dynamics produce patterns

resembling muscle activity. Conversely, the non-rotational patterns in SMA co-vary with cues

regarding when movement should be initiated. Thus, while SMA and motor cortex display

superficially similar single-neuron responses during visually guided reaching, their different

population dynamics indicate they are likely performing quite different computations.

DOI: 10.1038/s41467-018-05146-z OPEN

1 Department of Neuroscience, Columbia University Medical Center, New York, NY 10032, USA. 2Department of Statistics, Columbia University, New York,NY 10027, USA. 3 Zuckerman Mind Brain Behavior Institute, Columbia University, New York, NY 10027, USA. 4Grossman Center for the Statistics of Mind,Columbia University, New York, NY 10027, USA. 5 Center for Theoretical Neuroscience, Columbia University Medical Center, New York, NY 10032, USA.6 Kavli Institute for Brain Science, Columbia University Medical Center, New York, NY 10032, USA. Correspondence and requests for materials should beaddressed to M.M.C. (email: [email protected])

NATURE COMMUNICATIONS | (2018) 9:2754 | DOI: 10.1038/s41467-018-05146-z | www.nature.com/naturecommunications 1

1234

5678

90():,;

http://orcid.org/0000-0001-9125-3027

http://orcid.org/0000-0001-9125-3027

http://orcid.org/0000-0001-9125-3027

http://orcid.org/0000-0001-9125-3027

http://orcid.org/0000-0001-9125-3027

mailto:[email protected]

www.nature.com/naturecommunications


A basic goal of motor physiology is to characterize corticalresponses during voluntary movement. Characterizationrequires identifying parameters with which neural activity

covaries: e.g., muscle activation, hand velocity, or higher-orderfeatures1–3. Yet neural activity contains structure beyond thatreflecting movement parameters4–10, including structure reflect-ing dynamics that generate and control movement6,7,11–14. Herewe consider the supplementary motor area (SMA) and motorcortex, and ask whether they obey similar or different population-level dynamics. It is hypothesized that SMA is specialized forsequences and/or internally initiated movements15–25. SMA isnevertheless active during single movements26, including exter-nally prompted movements27–32. The relative roles of SMA andmotor cortex during non-sequential movements thus remainuncertain (for review see ref. 17). There exist at least three pos-sibilities. First, SMA and motor cortex may perform redundantcomputations. By analogy, the frontal eye fields and superiorcolliculus have different specializations, but respond similarlyduring simple saccades33. Second, SMA and motor cortex mayprocess different kinds of information, yet still perform compu-tations subserved by similar dynamics. This could be consistentwith the proposal that different cortical areas perform a canonicaloperation on different inputs34,35. Third, SMA and motor cortexmay perform very different computations via different classes ofdynamics.

Known differences between SMA and motor cortex do notresolve these possibilities. Anatomy suggests parallel com-plementary contributions36, but that is consistent with eitherdifferent or similar computations. In support of a higher-orderrole for SMA, SMA facilitates muscle activity less strongly37, isless responsive to proprioceptive input26,38, is more responsiveduring ipsilateral arm movements26, displays signals reflectingmovement outcome39, and contributes to learning betweenmovements40. Yet during standard visually guided movements,aspects of SMA and motor cortex responses are extremely similar,suggesting that they “operate in parallel”21 and may make largelyredundant contributions.

We addressed the relative contributions of SMA and motorcortex by examining the population-level dynamics that pre-sumably subserve network computations. This comparison isaided by recent characterizations of motor cortex dynamics.During reaching, the motor cortex population displays a “centralmotif” composed of two aspects: a condition-invariant shift instate5 immediately followed by state trajectories following rota-tional dynamics6,12,41. This same motif is naturally displayed bynetwork models trained to generate muscle activity patterns42.For such models, the central motif reflects the underlying com-putation. The condition-invariant shift initiates movement bybringing the state to a region where rotational dynamics dom-inate. The resulting oscillatory patterns form a basis for multi-phasic muscle commands. Rotational dynamics, related to quasi-periodic sub-movements, have also been observed in LFP43.Given the proposed connections between dynamics and function,the central motif represents a natural point of comparison.

We recorded neural responses from SMA and motor cortexwhile monkeys executed reaches to radially arranged targets.When examined using the population vector and populationPSTH, SMA and motor cortex showed similar structure, includ-ing preparatory and movement-related activity that covaried withreach direction. To examine dynamics, we employed a novel“hypothesis-guided” dimensionality reduction (HDR) approachthat translates a hypothesis into a cost function. Our cost functionsought projections where some dimensions capture a condition-invariant shift in state while other dimensions capture trajectoriesdescribed by generic linear dynamics.

Both SMA and motor cortex displayed a large, similarlyorganized, condition-invariant shift in population state justbefore movement initiation. Thus, the first aspect of the centralmotif was almost perfectly shared between the two areas, possiblyconstituting a shared signal related to movement initiation5. Yetin terms of dynamics, SMA and motor cortex were quite different.SMA activity was not well described by linear dynamics, dis-played weak rotational structure overall, and lacked the 1.5–3 Hzrotations previously reported in motor cortex. In contrast, HDRidentified multiple dimensions where motor cortex activityobeyed approximately linear dynamics. Motor cortex dynamicswere dominated by rotations, even though HDR did not speci-fically seek rotations. Rotations occurred in the 1.5–3 Hz rangeand produced response features matching multiphasic aspects ofmuscle activity. Although SMA lacked the clear dynamicalstructure found in motor cortex, it contained a complementarytype of information: SMA activity co-varied with the “higher-level” task constraints that determined when movement could beinitiated.

In summary, only in SMA did activity vary strongly withhigher-level task requirements. SMA and motor cortex bothshared a large signal previously shown to be temporally locked tomovement initiation. Finally, only motor cortex showed strongrotational dynamics. These different dynamics, and the differenttypes of information carried by neural activity, argue that SMAand motor cortex are performing very different computations.

ResultsTask. Two monkeys (Ba and Ax) executed radial reaches in eightdirections across three contexts: cue-initiated, self-initiated, andquasi-automatic. These contexts differed in the task requirementsgoverning how and when movement should be initiated. The cue-initiated context employed the standard instructed-delay para-digm: a randomized delay period (0–1000ms) separated targetonset from an explicit go cue. In the self-initiated context,monkeys were free to reach upon target presentation, but waitinglonger yielded larger rewards up to a limit at 1200 ms. The quasi-automatic context was similar to the cue-initiated context, but thego cue was the onset of target motion along a radial path towardthe screen’s edge. This context evoked low-latency reaches thatintercepted the target mid-flight. Target and central touch-pointcolor (red, blue, or yellow) cued the context. Trials were inter-leaved. Reaches had similar trajectories across contexts (Fig. 1a)but tended to be slightly faster for the quasi-automatic context(Fig. 1b, yellow).

In a separate study44, we exploit these contexts to examinepreparatory neural events in motor cortex. In the present study,we compare movement-related dynamics between areas. Giventhis goal, the advantage of different contexts is that they elicitresponses across a greater range of situations—including situa-tions that may differentially engage SMA. We analyzed only trialswith sufficient time, between target and movement onset, forclear preparatory activity to be established. This allowed analysisto concentrate on movement-related dynamics. For cue-initiatedand quasi-automatic contexts, we analyzed trials with delays>400 ms. For the self-initiated context, the monkeys’ behaviorprovided the desired separation.

Neural and muscle recordings. We recorded neural responsesfrom SMA (141 and 186 neurons for monkey Ba and Ax) andmotor cortex (129 and 172 neurons). Recordings used singleelectrodes or 24-channel linear electrode arrays. Recordings weremade from regions where electrical stimulation produced armmovements. Figure 1c illustrates where electrode penetrations

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05146-z

2 NATURE COMMUNICATIONS | (2018) 9:2754 | DOI: 10.1038/s41467-018-05146-z | www.nature.com/naturecommunications


entered (Supplementary Fig. 1 shows structural MRI). For SMA,nearly all recordings were made relatively deep, from the medialwall. For motor cortex, we recorded from sulcal primary motorcortex, surface primary motor cortex and the immediately adja-cent aspect of caudal PMd. Across these recordings we observedthe expected gradient of stronger preparatory activity on thesurface versus sulcus. Yet this tendency was far from completeand responses formed a continuum with no noticeable dis-continuity with location. We thus analyzed primary and pre-motor cortex recordings together as a single motor cortexpopulation. This is consistent with our prior finding that primaryand premotor cortex display similar dynamics when analyzedseparately or together6.

Spike-trains during a target-locked epoch and a movement-locked epoch were concatenated (Fig. 1d, e; gray circles indicateconcatenation time) allowing computation of an across-trial firingrate that spans both events with a representative separation(traces at bottom of each panel). This unified rate was useful forvisualizing preparatory and movement-related events together.However, all analyses of dynamics focus on the movement-aligned epoch, after the time of concatenation. Muscle activitywas recorded percutaneously from the major muscles of theupper arm (13 and 10 recordings for monkey Ba and Ax). Weemployed the standard technique of rectifying the voltage traces,which provides a net measure of the activity of many motor units.The average response of a given muscle for a given condition

100 ms

Monkey Ba Monkey Ax

Motorcortex

SMA

SMA

0

30

Firi

ng r

ate

(Hz)

t100 ms

21

0

Tria

lF

iring

rat

e (H

z)T

rial

Motorcortex

0

80

t m100 ms

16

0

EMG

0

40

Rec

t. vo

ltage

(A.U

.)

t m100 ms

22

0

Tria

l

a d

b e

c f

m

*

Fig. 1 Illustration of behavior and physiological recordings. a Average reach trajectories for the eight directions and three contexts. Red, blue and yellowtraces show average hand position during curing cue-initiated, self-initiated, and quasi-automatic reaches. Traces are shaded from dark to light based onreach direction. Red traces are dashed to allow visualization of blue traces with which they often overlap. Data shown are for monkey Ba, and were similarfor monkey Ax. Star indicates the reach direction for which neural/muscle data are shown in panel (d). Scale bar shows 2 cm. b Average hand-velocityprofiles corresponding to the trajectories in panel (a). Scale bar shows 1 m/s. c Reconstructions of surface landmarks based on MRIs (see SupplementaryFigure 1 for example MRI sections). Shaded regions indicate where penetrations entered cortex, and are shaded darker to indicate where recordingsincluded deeper locations. Scale bar shows 5mm. d Raster plot of spikes recorded from one SMA neuron for one reach direction during the quasi-automatic condition. Data in this and subsequent panels are for monkey Ax. Data to the left of the gray symbol are aligned to target onset (t) and data tothe right are aligned to movement onset (m). Filtering and averaging of spike-trains yields a smooth firing rate versus time (black trace) that interpolatesacross the concatenation at the time indicated by the gray symbol. Filtering used a narrow (20ms) Gaussian to ensure high-frequency aspects of theresponse were not lost. e Same as in d but for a neuron recorded from motor cortex. f Similar to d and e but for EMG recorded from the medial deltoid.Voltage traces show a mixture of discrete and (especially during movement) overlapping events. Discontinuities resulting from concatenation of target-locked and movement-locked data are small and barely visible. Rectification, filtering and averaging produces a continuous trace summarizing averageEMG intensity (magenta trace)

NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05146-z ARTICLE




(Fig. 1f, magenta trace) was then computed just as for theneurons.

On average, firing rates were higher in motor cortex: peakfiring rate averaged 75 and 77 spikes/s (monkey Ba and Ax)versus 43 and 53 spikes/s for SMA. Individual-neuron firing rate

estimates thus tended to be slightly noisier for SMA. Otherwise,single-neuron responses were in many ways similar (Fig. 2a, b).For both areas, firing rates varied with reach direction before andduring movement (darker/lighter traces correspond to leftwards/rightwards reaches). During movement, responses often exhibited

Target Move

Ax 082516

Ax 093016

Ax 080216

Ax 093016

Ba 052615

Ba 041214

Ax 082814

Ba 042814

Ax 042116

Ba 121713

Ba anterior deltoid

Ba medial deltoid

Ba pectoralis

Ax trapezius

100 ms

Ax medial deltoid

a b cMotorcortex EMGSMA

Monkey Ba Monkey Axd e

0 2 4 6 8 100

0.5

0 2 4 6 8 10Frequency (Hz)

0

0.5

Nor

mal

ized

am

plitu

de

Frequency (Hz)

Nor

mal

ized

am

plitu

de

Motorcortex

EMG

SMA

Fig. 2 Example responses of single neurons and muscles. a Example responses of five neurons recorded from SMA. Each colored trace plots the trial-averaged firing rate for one condition, computed as illustrated in Fig. 1. Red, blue, and yellow traces correspond to the three contexts. Darker/lighter tracesare for reaches to the left/right. (Same color scheme as in Fig. 1a, b). Scale bars indicate 20 spikes/s. b Same as a, but for five neurons recorded frommotor cortex. c Same as a and b, but for five example muscle recordings. The bottom of the vertical scale bars indicates zero EMG activity, but the scale isotherwise arbitrary. d Frequency spectrum, computed via the Fourier transform, for the three populations. Frequency content was computed per neuron/muscle, over the temporal interval from −250 to 250ms relative to movement onset. Frequency content was then normalized and averaged. Envelopesshow 95% confidence intervals computed via bootstrap, resampling neurons/muscles. Data are for monkey Ba. e Same as d, but for monkey Ax




multiple peaks and/or phases that varied with condition.Individual neurons displayed a variety of diverse and complexresponses45. Directional selectivity often differed before andduring movement, or early versus late during movement45–50.Robust activity was typically present across contexts. The aboveresults are consistent with the finding that SMA has directionalresponses not unlike those in motor cortex, and is responsiveduring both self-initiated and externally cued movements27,28,31.

Muscle responses were also often complex and multiphasic(Fig. 2c). Muscles did not show changes in activity until justbefore movement onset, with some exceptions. For example,baseline activity of the medial deltoid for monkey Ax (thirdsubpanel) increased slightly overall, well before movement (moreso for the quasi-automatic context). These changes presumablyreflect a slight tensing in anticipation of the pending reach.Relative to neural activity in the same time-range, changes inmuscle activity were sparse, small, and weakly directional.

Standard population analyses for SMA and motor cortex. Thepopulation vector51, a summary based on directional aspects ofactivity, behaved similarly for SMA and motor cortex (Fig. 3a, b,monkey Ba; Supplementary Fig. 2, monkey Ax). We found onlysmall differences between the two areas. We first considered thevector magnitude, relative to the firing rates of the contributing

neurons. Comparing SMA with motor cortex, vector magnitudewas similar: slightly larger for monkey Ba (5%, N.S.) and smallerfor monkey Ax (24%, p < 0.01). For both monkeys, the SMApopulation vector was slightly, but not significantly, less-wellaligned with target direction.

We also employed the population PSTH: for each neuron, weidentified the most-preferred condition (the condition that evokedthe largest firing rate over a 500ms window centered onmovement onset) then averaged that response across neurons. Asimilar average was produced for the least-preferred condition andall intermediate conditions. This analysis was repeated separatelyfor each context. The resulting population PSTHs (Fig. 3c, d)reveal that preparatory tuning (variation of firing rate with targetdirection) developed shortly after target onset for the cue-initiatedand quasi-automatic contexts (red and yellow) and somewhat laterfor the self-initiated context (blue). Stronger movement-relatedtuning then arose ~150 ms before movement onset.

Population PSTHs showed only modest differences betweenareas. Population PSTHs reveal slightly stronger preparatoryversus movement-period tuning for SMA. This was consistentwith single-neuron observations; the median ratio of preparatoryto movement tuning was higher for SMA versus motor cortex(0.79 versus 0.52 for monkey Ba; 0.81 versus 0.64 for monkey Ax;p < 0.01 for both via rank sum test). For monkey Ba tuning tended

Target Move Target MoveTarget Move20 s

pike

s/s

100 ms

Cue-initiated

Self-initiated

Quasi-automatic

Target Move Target MoveTarget Move

a

b

c

d

Motorcortex

SMA

Motorcortex

SMA

Fig. 3 Population vectors and population PSTHs for monkey Ba. a SMA population vectors for each of the three contexts. Each cluster of lines correspondsto one of the eight reach directions. The length of each black line indicates the firing rate of one neuron for that direction and context, and points in thatneuron’s preferred direction. Colored traces plot (on a different scale) the sum of those vectors. For each neuron, the preferred direction was computedbased on firing rates across all directions and contexts. b Motor cortex population vectors. Scaling is arbitrary and differs between the two brain areas, asthey had different average firing rates (see main text). c SMA population PSTHs for the three contexts. Shading is ordered from most-preferred direction(lightest) to least-preferred (darkest) within each context. d Motor cortex population PSTHs





to remain high slightly longer for SMA, but the opposite tendencywas observed for monkey Ax. Despite these modest differences,population PSTHs were strongly correlated between areas. Formonkey Ba, correlations were 0.92, 0.96, and 0.92 for the threecontexts. For monkey Ax, correlations were 0.94, 0.96, and 0.92.The lower bound of the 95% confidence intervals was >0.90 for allcomparisons.

Frequency spectra also revealed only modest differencesbetween SMA and motor cortex (Fig. 2d, e). There was slightlygreater power in the 1.5–3 Hz range for motor cortex (and for themuscles) versus SMA. This range includes frequencies followedby rotational dynamics in prior studies. This is potentiallysuggestive, but no more: SMA also had considerable power in thisrange.

Thus, SMA and motor cortex appear reasonably similar whenanalyzed via these standard methods. However, these methods aresuited to revealing certain aspects of the population response butnot others. For example, the frequency spectrum does notdistinguish between power organized into coherent rotationsversus disorganized trajectories. The population vector confirmsthe presence of directional responses but does not capture otherresponse properties. Population PSTHs reveal the rough envelopeof tuning, but not how activity evolves within that envelope.Indeed, population PSTHs incorrectly suggests that tuningremains consistent with time. Analyses of dynamics provide analternative approach that might sidestep these limitations.

Hypothesis-guided dimensionality reduction. We employ theperspective that single-neuron responses reflect latent variablesthat are shared across the population11,52–57 and can be estimatedusing dimensionality reduction methods58–63. A standard linearmodel of each neuron’s response is:

at;n ¼Xk

xt;kwk;n ð1Þ

where at,n is the firing rate of neuron n at time t, xt,k is the value ofthe kth latent variable, and wk,n determines the contribution ofthat latent variable to the response of neuron n. A key question iswhether the evolution of xt, the vector of latent variables, isdescribed by a dynamical flow field. Is it the case that _xt � f xtð Þfor some function f(·), perhaps with some linear approximation:_xt � xtD? If so, is D dominated by rotations or other forms ofdynamics? These questions cannot be addressed at the single-neuron level. Neither multiphasic firing rates6 nor data smooth-ness imply rotational dynamics64; much depends on how phasesare coordinated across neurons and conditions65.

We previously examined motor cortex responses using amethod, jPCA, that seeks latent variables described by rotationaldynamics. jPCA has two shortcomings given our present goals.First, when comparing areas, we wish to make fewer assumptionsregarding the form of dynamics. Second, the central motifpredicted by motor-cortex network models includes bothrotational dynamics and a condition-invariant shift of the neuralstate42. We previously resorted to multiple methods (dPCA66

followed by jPCA) to test for the presence of the central motif5.That approach is suboptimal; latent variables should ideally befound in a unified fashion.

To do so, we employ a hypothesis-guided dimensionalityreduction (HDR) methodology. Recent work observes that mostdimensionality reduction methods implicitly or explicitly employa cost function67, and that different cost functions embodydifferent hypotheses. For example, PCA embodies the simplehypothesis that the most relevant signals are the largest signals,while dPCA66 posits that different dimensions contain activitythat co-varies with different task parameters. Here we follow this

lead and leverage the suggestion that “future linear dimension-ality reduction algorithms can be derived in a simpler and moreprincipled fashion”67. We adopt a cost function tailored to ourspecific hypothesis: there may exist a projection of the data thatcaptures a large percentage of response variance, with somedimensions capturing condition-invariant structure and otherdimensions capturing structure described by linear dynamics.

We consider the data matrix, A, where each column containsthe firing rate of one neuron across times and conditions. Weestimate the latent variables as a projection, X=AWT, where eachcolumn of X contains the values of one latent variable acrosstimes and conditions. The rows of the orthogonal matrix W arethe “neural dimensions”, found by minimizing a cost function f(W). Because the above hypothesis contains three components,we employ a tripartite cost function:

f Wð Þ ¼ frec Wð Þ þ finvar Wð Þ þ fdyn Wð Þ ð2Þ

The first term, frec(W), is identical to the PCA cost function, andis small if the latent variables capture considerable variance (i.e., ifthe firing rates in A are accurately reconstructed by Arec=XW).The second two terms relate to the hypothesis that there existscondition-invariant structure in some dimensions and dynamicalstructure in other, orthogonal dimensions. If so, an appropriateW will result in latent variables X=[Xinvar, Xdyn]. Xinvar are latentvariables that vary with time but not condition. Xdyn are latentvariables whose evolution obeys linear dynamics. finvar(W) issmall if Xinvar varies strongly with time but not condition. fdyn(W)is small if the fit _Xdyn � XdynD is good for some choice of D.Equations for frec(W), finvar(W), and fdyn(W) are provided in theMethods. By minimizing f(W) we ask the question: does thereexist a projection of the population response that capturesconsiderable variance and has the hypothesized condition-invariant and dynamical structure?

For each population, we minimized f(W) via gradient descentonW, then used those dimensions to find the latent variables. Werefer to Xinvar as the projection onto “condition-invariantdimensions”. Of course, whether Xinvar actually displays thehypothesized condition-invariant structure is an empiricalquestion. We refer to Xdyn as the projection onto “dynamicaldimensions”. Again, whether Xdyn actually displays dynamicalstructure is an empirical question. We set the total number ofdimensions to six, and sought two condition-invariant dimen-sions and four dynamical dimensions.

As with other methods, it is important to know whetherdimensions capture considerable data variance. Notably, PCAminimizes frec(W) while we are asking HDR to minimizefrec(W) + finvar(W) + fdyn(W). Thus, the variance captured byPCA is the maximum that could possibly be captured by HDR;HDR must sacrifice some captured variance as it seeks thehypothesized structure. In practice, the six HDR dimensionscaptured only modestly less variance than the first six PCs, and atleast as much variance as PCs 2-7. This was true for bothmonkeys, both cortical areas, and the muscle populations(Supplementary Fig. 3). Employing more than six dimensionscaptured slightly more variance but yielded little improvement incapturing condition-invariant or dynamical structure.

HDR optimizes jointly for all aspects of the hypothesizedstructure. In contrast jPCA employs PCA or dPCA and thenseeks rotational structure5,6, which could cause structure to bemissed. Unlike jPCA, the present use of HDR does not focus onrotations per se, reducing concerns that the method imposes aparticular form of dynamics. HDR is thus simultaneously moreprincipled, more powerful, and more conservative that pastapproaches.




Qualitative assessment of dynamical structure. We plotted pairsof latent variables against one another, yielding projections of thedata onto two-dimensional planes. Figure 4 shows three suchplanes for monkey Ba. We first focus on two “dynamical planes”,each showing the projection onto two of four dynamicaldimensions. Within a projection, each colored state trajectorydescribes how those latent variables evolved over time for onecondition (one direction/context). For motor cortex (Fig. 4,middle row) both dynamical planes captured rotational structure.Trajectories rotated in the same direction, at approximately thesame angular velocity, with phase and amplitude varying acrossconditions. Trajectories are shown from 80ms before until 150ms after movement onset (reaches lasted ~200 ms). Trajectoriescontinued rotating for 50–100 ms after the interval shown (theshorter plotted interval minimizes overlapping traces).

The SMA population (Fig. 4, top row) did not exhibit structurethat followed a clear dynamical flow-field (quantification tofollow). Rotational structure was weak, and trajectories appearedsomewhat disorganized. We stress that this does not imply thatthe SMA population response is truly disorganized, simply that itis not well described by the hypothesis of approximately lineardynamics. As will be described subsequently, SMA did exhibitother clear structure that could be identified by the HDRapproach.

Individual-muscle responses were often multi-phasic, and inmany ways resembled individual-neuron responses. Yet the

muscle population showed little coherent rotational tendency(Fig. 4, bottom row). State trajectories often formed loops, but thesign and degree of curvature was inconsistent across conditions.Still, visual inspection suggests some potential commonalitybetween muscle and motor-cortex populations. For example, thefirst dynamical plane for the muscles resembled that for motorcortex, but viewed from the side. We will return below to thispotential connection.

Results were similar for monkey Ax (Fig. 5). The motor cortexpopulation showed robust rotations in two dynamical planes.Some weak rotational structure was present for SMA (more sothan for monkey Ba) but the flow-field was not well organized:some trajectories clearly rotated, but many others did not. Themuscle population also showed little dynamical structure; loopingtrajectories were not organized into coherent rotations.

Quantification of dynamical structure. For each population, weasked how well Xdyn, the state in the dynamical dimensions, obeyslinear dynamics. We fit with _Xdyn � XdynD, where _Xdyn is thetime derivative of Xdyn, and D is the matrix that provides the bestfit. D is unconstrained and can capture rotational or other lineardynamics. Results are summarized in Fig. 6a, b (bars labeled “D”).For SMA, the dynamical fit was poor (R2= 0.22, monkey Ba) ormoderate (R2= 0.52, monkey Ax). The dynamical fit wasalso moderate for the muscle populations: R2= 0.41 and

dyn 2

dyn1 dyn4 Invar1

dyn1 dyn4 Invar1

dyn1 dyn4 Invar1

dyn 3

dyn 1

dyn 2

dyn 3

dyn 1

dyn 2

dyn 3

dyn 1

First dynamicalplane

Second dynamicalplane

Dynamical vscondition-invariant

Motorcortex

EMG

SMA

Fig. 4 State-space plots of the latent variables identified by HDR for monkey Ba. HDR was applied separately for SMA, motor cortex, and musclepopulations. Each sub-panel plots the joint evolution of two latent variables (equivalent to a projection of the full neural state onto two dimensions). Eachcolored trajectory corresponds to one condition and plots the values of the latent variables from −80 to 150ms relative to movement onset. The beginningand end of each trajectory is indicated by a circle and arrow, respectively. Color-coding is as in Figs. 1a, b and 2a–c: red, blue, and yellow trajectoriescorrespond to the three contexts, while darker/lighter traces correspond to leftward/rightward movements. Dashed lines are included to aid visualizationand indicate the local direction of flow (this was often less consistent for SMA and the muscles than for motor cortex). Scaling is arbitrary but the samescale is always used for the horizontal and vertical axis within each subpanel





0.46. The dynamical fit was best for the motor cortex populations:R2= 0.84 and 0.76. The difference between SMA and motorcortex was significant (p < 0.001 for both monkeys) based on abootstrap that redrew conditions to establish confidence intervals.We also employed a highly conservative bootstrap that redrewdimensions rather than conditions (Methods). This approachtreats the combined SMA and motor cortex latent variables as alarge undifferentiated set of latent variables in a common popu-lation. The bootstrap then asks how often two arbitrary “areas”,each containing a subset of those variables, differ by as much asthe empirical SMA and motor cortex datasets. Even via this veryconservative method, the difference between SMA and motorcortex was significant: p < 0.0001 for monkey Ba and p < 0.04 formonkey Ax.

The projections in Figs. 4 and 5 suggest that a large differencebetween SMA and motor cortex dynamics is the prevalence ofrotations. To quantify this, we decomposed the matrix D into itssymmetric and skew-symmetric components D=Dsym+Dskew. Ifdynamics are dominated by rotations, D will be naturally close toskew symmetric, such that Dskew≈D. If so, the dynamical fitprovided by Dskew will be both high and nearly as good as thedynamical fit provided by D.

Motor cortex dynamics were dominated by rotations in a waythat SMA dynamics were not. For monkey Ba, the fit provided byDskew was more than six-fold better for motor cortex versus SMA(Fig. 6a, compare middle blue and black bars; R2= 0.74 versus0.11). For monkey Ax, the fit provided by Dskew was almost three-fold better for motor cortex versus SMA (Fig. 6b, compare middleblue and black bars; R2= 0.62 versus 0.23). These differenceswere statistically significant (for the conservative test: p < 0.001for monkey Ba and p < 0.05 for monkey Ax). The different

dominance of rotational dynamics is also apparent whencomparing within each area. For SMA, the R2 associated withDskew was at most half as large the R2 associated with D (Fig. 6a,b, compare middle and left black bars). For motor cortex the R2

associated with Dskew was almost as high as the R2 associated withD (Fig. 6a, b, compare middle and left blue bars). For the muscles,the R2 associated with Dskew was negative (Fig. 6a, b, middlemagenta bar) and significantly different from that in motor cortexby both tests. (Negative values of R2 are possible if D is far fromskew-symmetric).

In addition to considering Dskew (the skew-symmetriccomponent of the best-fit dynamics matrix), we also consideredD�

skew, the skew-symmetric matrix that provides the best fit. ForSMA, the R2 associated with D�

skew was low (Fig. 6a, b, rightblack bar). For motor cortex, the R2 associated with D�

skew washigher (Fig. 6a, b, right blue bar) and only modestly less thanthe R2 associated with D. Comparing SMA versus motor cortex,the R2 associated with D�

skew was statistically different viabootstrap (for the conservative test: p < 0.001 for monkey Baand p= 0.01 for monkey Ax). For the muscles, the R2

associated with D�skew was small and statistically different from

that for motor cortex (for the conservative test: p < 0.0001 forboth monkeys).

As suggested by inspection of Figs. 4 and 5, what littlerotational structure was present in SMA occurred at frequencieslower than in motor cortex. To quantify frequency, we analyzedthe purely rotational system described by D�

skew. For SMA, allrotational frequencies were below 1.1 Hz for both monkeys andboth dynamical planes (Fig. 6c, d, black bars). For motor cortex,rotational frequencies were higher: the greater was ~3 Hz and thelesser was between 1.5 and 2 Hz (Fig. 6c, d, blue bars). The 1.5–3

First dynamicalplane

Second dynamicalplane

Dynamical vscondition-invariant

Motorcortex

EMG

SMA

dyn 2

dyn 3

dyn 1

dyn 2

dyn 3

dyn 1

dyn 2

dyn 3

dyn 1

dyn1 dyn4 Invar1

dyn1 dyn4 Invar1

dyn1 dyn4 Invar1

Fig. 5 State-space plots for monkey Ax. Format as in Fig. 4. Scaling is arbitrary but the same scale is always used for the horizontal and vertical axis withineach subpanel. Scaling is sometimes altered across subpanels. In particular, for motor cortex, scaling was reduced for the third column to allow plotting ofthe first condition-independent latent variable, which had a very large magnitude




Hz frequencies in motor cortex are consistent with prior findings,and with models that use oscillatory dynamics to provide a basis-set for outgoing muscle-commands6,41,42.

Thus, SMA and motor cortex differed in essentially everyaspect of their dynamical structure. SMA population activity wasless well fit by linear dynamics overall. SMA dynamics were notdominated by rotations, and what rotational structure waspresent occurred at frequencies lower than in motor cortex. Apotential concern is that perhaps SMA and motor cortex are trulysimilar, but dynamics in SMA were missed because lower firingrates yielded lower signal-to-noise. This is unlikely for threereasons. First, dimensionality reduction effectively denoisesresponses by leveraging commonalities across neurons68. Giventhat we recorded 141 (monkey Ba) and 186 (monkey Ax) neuronsfrom SMA, signal-to-noise is unlikely to be a large issue. Second,if signal-to-noise were poor in our SMA recordings, the variancecaptured by HDR should be much lower for SMA. In fact, bothHDR and PCA captured slightly more variance for SMA versusmotor cortex (Supplementary Fig. 3). Third, condition-invariantstructure would be weakened if noise dominated signal. Yet aswill be described below, HDR readily identified condition-invariant structure for both SMA and motor cortex.

Qualitative assessment of condition-invariant structure. HDRrevealed a condition-invariant shift in state for both areas. Thethird column of Figs. 4 and 5 shows the projection onto the firstdynamical dimension and the first condition-invariant dimen-sion. For motor cortex, the resulting two-dimensional view can be

thought of as taking the first dynamical plane and spinning it soas to view the rotations “on edge”, revealing new structure in athird dimension. This view reveals a large condition-invarianttranslation: trajectories start at the left for every condition, andtranslate a roughly equal distance to the right. Following thetranslation, rotations (viewed on edge) occur on the right side ofthe plane. This structure agrees with prior findings in motorcortex and in simulated networks5,42.

We found that SMA also displayed a condition-invarianttranslation of the neural state, something not previously reported.This condition-invariant translation occurred alongside selectivityfor condition in other dimensions (e.g., the first dynamicaldimension separated trajectories across conditions). Both SMAand motor cortex also showed a second dimension withcondition-invariant structure (see subsequent analyses).

Quantification of condition-invariant structure. We assessedthe degree to which projections onto the condition-invariantdimensions were truly condition invariant. To quantify “condi-tion invariance”, we divided the variance of the across-conditionmean by the total variance across all times and conditions. If asignal is identical across conditions, condition invariance will be100%. Conversely, a signal that varies strongly with condition canhave condition invariance approaching 0%. Both SMA and motorcortex contained relatively pure condition-invariant structure(Fig. 7a, b). For SMA, condition invariance was 96% and 95%(monkey Ba and Ax). For motor cortex, condition-invariance was

D D skew Fre

q 1

Freq 2

Freq 1

Freq 2

0

4

D skew

0

1

D

a c

Hz

Dynamical / rotational fit Rotation frequency

R2

R2

0

1

Hz

0

3

b d

SMA Motorcortex

EMG

*D sk

ew *D sk

ew

D D skew *

D skew

D D skew Fre

q 1

Freq 2

Freq 1

Freq 2

D skew

D *D sk

ew *D sk

ew

D D skew *

D skew

SMAMotorcortex

Fig. 6 Quantification of dynamical structure. Insets indicate the nature of the structure being quantified. a Fit quality provided by unconstrained (D) andconstrained (Dskew and D�

skew) linear dynamical fits. Fits are made over a 200ms window, starting at movement onset, during which firing rates arechanging rapidly. Results were robust to reasonable window sizes, so long as they included the time when responses were in rapid flux. For example, all p-values quoted in the text are lower (more significant) when using an earlier window: from 80ms before movement onset until 150ms after. Negativevalues, in the case of EMG, have been truncated. Data are for monkey Ba. b Same as a but for monkey Ax. c Rotational frequencies derived from D�

skew forSMA and motor cortex. Data are for monkey Ba. d Same as c but for monkey Ax





97% and 98%. For the muscles, condition invariance was mod-estly lower: 86 and 91%.

For both SMA and motor cortex, the condition-invariant signalwas the dominant signal in the data: the two condition-invariantdimensions captured nearly as much variance as the first two PCs(Fig. 7a, b). For the muscles, condition-invariant structure wasmuch weaker; the condition-invariant dimensions captured farless variance than the first two PCs. Thus, a condition-invariantsignal is prevalent only in SMA and motor cortex. Subsequentanalyses will explore whether the structure of that signal is similarin both areas.

HDR-independent quantification of dynamical structure. Bydesign, the finvar(W) and fdyn(W) terms compete—any structure

captured by condition-invariant dimensions cannot be capturedby dynamical dimensions. Might this cause HDR to miss dyna-mical structure, perhaps exaggerating the difference betweenSMA and motor cortex? Empirically this was not the case.Reducing the weighting of finvar(W) by a factor of ten had littleeffect: dynamical fit quality changed little and the same differencepersisted between SMA and motor cortex.

We also applied jPCA, which is more aggressive in seekingrotational structure. This becomes advantageous if there is aconcern that rotational structure might be missed. jPCA revealeddifferences between SMA and motor cortex similar to thoserevealed by HDR. For monkey Ba, jPCA yielded a dynamical fitwith an R2 of 0.12 ± 0.02 for SMA versus 0.59 ± 0.05 for motorcortex (SEMs via bootstrap resampling neurons). For monkey Ax,

a

PCs 1:2

50

100

Inva

rianc

e0

0.5

Var

. cap

t.

Inva

r. 1:

2

Inva

r. 1:

2

PCs 1:2

50

100

0

0.5

Inva

r. 1:

2

Inva

r. 1:

2

Inva

rianc

e

Var

. cap

t.

b

0

0.5

PCs 1:2

50

100V

ar. c

apt.

Inva

rianc

e

Inva

r. 1:

2

Inva

r. 1:

20

0.5

PCs 1:2

50

100

Inva

r. 1:

2

Inva

r. 1:

2

Var

. cap

t.

Inva

rianc

e

0

0.5

PCs 1:2

50

100

Inva

r. 1:

2

Inva

r. 1:

2

Var

. cap

t.

Inva

rianc

e

50

100

0

0.5

Inva

r. 1:

2

Inva

r. 1:

2

Inva

rianc

e

Var

. cap

t.

PCs 1:2

SMA Motorcortex

EMG

–0.05 0.05

0.2i 0.2i 0.2i 0.2i

–0.2i –0.2i –0.2i –0.2i

–0.05 0.05 –0.05 0.05 –0.05 0.05

SMAMotorcortex

dc

Fig. 7 Quantification of HDR-based condition-invariant structure and PCA-based dynamical structure. a Quantification of the structure captured by thecondition-invariant dimensions. In each subpanel, the leftmost bar indicates the empirical condition invariance of activity in the condition-invariantdimensions. The right plot indicates the total data variance captured by the two condition-invariant dimensions and that captured by the first two PCs (graybar). These analyses focus on the time from −100ms before movement onset until 200ms after movement onset. This is the epoch during which thesignal in the condition-invariant dimensions was changing rapidly. Data are for monkey Ba. b Same as a but for monkey Ax. c Eigenvalue spectra whenfitting linear dynamics to the population response projected onto the top six principal components (PCs), rather than using HDR. The analysis wasconducted for 100 bootstrap repetitions, each time re-drawing a new population of neurons (with replacement) from the recorded population. Fitting lineardynamics resulted in a set of six eigenvalues per bootstrap repetition. For visualization, only the first 50 sets of eigenvalues are shown. Circle sizes areproportional to fit quality. Data are for monkey Ba. d. Same analysis for monkey Ax




the corresponding R2 was 0.28 ± 0.05 for SMA versus 0.55 ± 0.04for motor cortex. As with HDR, rotational frequencies differedacross areas. For monkey Ba, the highest rotational frequency was0.65 ± 0.13 Hz for SMA versus 2.18 ± 0.25 Hz for motor cortex.For monkey Ax, the highest rotational frequency was 0.95 ± 0.19Hz for SMA versus 2.19 ± 0.32 Hz for motor cortex.

Finally, we applied a less principled but very simple approach.We fit an unconstrained linear dynamical system to trajectories inthe six dimensions, found via PCA, that captured the mostcondition-specific variance. Analysis was repeated, drawing newpopulations to allow bootstrap-based confidence intervals.Figure 7c, d plots the resulting eigenvalue spectra (each symbolplots one of the six eigenvalues for a given bootstrap repetition).Symbol size indicates fit quality, which was significantly lower forSMA versus motor cortex (monkey Ba: R2= 0.15 ± 0.02 versus0.63 ± 0.05; monkey Ax: R2=0.33 ± 0.05 versus 0.60 ± 0.04).Eigenvalue structure also differed. For SMA, the imaginary

(rotation-inducing) component never became as large as formotor cortex. For monkey Ba, the largest eigenvalue (of the non-bootstrapped data) had an imaginary component of 0.040i ±0.011i for SMA versus 0.144i ± 0.17i for motor cortex (0.144icorresponds to a frequency of 2.3 Hz). The corresponding valuesfor monkey Ax were 0.044i ± 0.017i versus 0.92i ± 0.18i. Formotor cortex, eigenvalues tended to form two clusters at distinctfrequencies. Little or no comparable structure was visible forSMA.

Potential functions of SMA and motor cortex signals. It hasbeen hypothesized that rotational dynamics in motor cortexrelate to the generation of multiphasic aspects of muscle activitywithin a movement6,41,42,65 or across sub-movements43. Toexplore a possible connection with muscle responses, weregressed muscle activity versus SMA and motor cortex activity.

SMA Motor cortex

M

M

SMA Motor cortex

0

20

% C

onte

xt-d

epen

dent

var

ianc

e

0

20

% C

onte

xt-d

epen

dent

var

ianc

e

Contributions to EMG fit (Ba)

Shared response aspects (Ba)

0 10

–0.05

0

0.05

Diff

eren

ce in

pow

er

–0.15

0

0.1

Diff

eren

ce in

pow

er

Frequency

0 10Frequency

Contributions toEMG fit (Ax)

Shared response aspects (Ax)

a c

b d

e

f

g

h

SM

A

Mot

or c

orte

x

EM

GFig. 8 Potential contributions to outputs and internal computations. a Relative contributions of SMA and motor cortex when fitting muscle activity withneural activity. Analysis is based on the four dynamical dimensions for SMA, motor cortex, and the muscles. Activity in each EMG dimension wasregressed against eight dimensions of neural activity, four each from SMA and motor cortex. The fit was the sum of contributions from SMA and motorcortex, shown separately in the left and right columns. Average R2 was 0.58. For visualization, contributions are shown for two of the four muscledimensions, chosen to highlight the greater contribution of motor cortex to multi-phasic structure. Each trace corresponds to one condition, plotted versustime over a 500ms period starting 150ms before movement onset (scale bar shows 100ms). Data are for monkey Ba. b Same analysis for monkey Ax. Theaverage R2 was 0.72. c Quantification of the effects illustrated in panel a. Difference in normalized power, between SMA and motor cortex contributions, asa function of frequency. Positive values indicate greater power in SMA, negative values indicate greater power in motor cortex. Averages are takenacross all conditions, and across all four dimensions of the EMG activity being fit. Flanking traces shown standard errors. d Same analysis for monkeyAx. e Percentage of total response variance due to activity varying with context. Data are for monkey Ba. f Same analysis for monkey Ax. g Responseaspects that were most strongly shared between SMA and motor cortex, found via canonical correlation analysis. Dimensions were found within the spacespanned by the six HDR dimensions, including the two condition-invariant dimensions and the four dynamical dimensions. Top and bottom rows plot thefirst and second canonical correlates. These are plotted versus time around the time of movement onset (M). Each trace corresponds to one condition.Scale bar shows 100ms. Data are for monkey Ba. h Same analysis for monkey Ax





This analysis considered only the dynamical dimensions, whichcaptured condition-specific response aspects. Each dimensionof muscle activity was simultaneously regressed against all eightdimensions of neural activity (four each from SMA and motorcortex). The full fit (average R2 of 0.58 and 0.72 for monkey Baand Ax) was thus the sum of contributions from SMA andmotor cortex (Fig. 8a, b). The SMA contribution was dominatedby slower signals; higher frequencies were present primarily inthe form of rapid ramps. In contrast, the motor cortex con-tribution contained overtly multiphasic structure (which dif-fered across monkeys, as did the muscle responses being fit). Toprovide quantification, we computed the difference in the fre-quency content of SMA and motor cortex contributions(Fig. 8c, d). SMA showed a relative lack of contribution in the3–4 Hz range for monkey Ba and the 2–4 Hz range for monkeyAx. Thus, the dynamical dimensions in SMA are less well-suited than those in motor cortex to contribute multi-phasicaspects of muscle activity.

Yet the dynamical dimensions in SMA contained a differenttype of structure: activity in the dynamical dimensions varied notonly with direction, but also with the three contexts. Thesecontexts varied regarding the rules and cues that determinedwhen movement should be initiated. Yet the physical reacheswere similar across contexts (Fig. 1a, b) as were both muscleactivity and movement-related motor cortex activity44. Thepresent analysis confirms prior findings: for motor cortex, nearlyall variance in the dynamical dimensions occurred across timeand/or reach direction. Very little (2–3%) occurred acrosscontexts (Fig. 8e, f). The same was true of the muscles. However,we found that context had a much larger influence on the neuralstate in the SMA dynamical dimensions.

These findings are consistent with the oft-proposed role ofSMA in movement initiation. A computation that determineswhen to move must consider the rules regarding when movementis allowed. However, other factors (e.g., visual cues and rewardprobability) also differed across contexts and could also beresponsible for the observed effects. What is clear is that SMAresponses co-vary with factors beyond the movements them-selves, in a way that muscle activity and motor cortex activity donot.

SMA and motor cortex share condition-invariant signals. Theabove analyses reveal that the SMA and motor cortex popula-tion responses differ in multiple ways. To ask whether therealso exist shared signals, we employed canonical correlationanalysis (CCA). CCA returns projections of one dataset that aresimilar to (correlate with) projections of another dataset. Weapplied CCA to the six dimensions returned by HDR andexamined the top two canonical variables; i.e., the most cor-related projections. Each canonical variable is plotted versustime (Fig. 8g, h) to allow comparison of temporal structure. Thetop two canonical variables were highly correlated betweenmotor cortex and SMA (r= 0.99 and 0.95 for monkey Ba; r=0.99 and 0.96 for monkey Ax) and were very close to conditioninvariant. The condition invariance of the first canonical vari-able was 97 and 98% (SMA and motor cortex) for monkey Baand 98 and 98% for monkey Ax. The condition invariance ofthe second canonical variable was 87 and 90% (SMA and motorcortex) for monkey Ba and 89 and 93% for monkey Ax.

The plots versus time in Fig. 8g, h are simply a complementaryapproach, relative to the state-space plots, of viewing condition-invariant structure. For example, the top row in Fig. 8g reveals alargely condition-invariant shift in state from low to high. Plottedin state-space this corresponds to a left-to-right shift in statecommon to all conditions, as in Fig. 4 (right column). Thus, both

SMA and motor cortex share a large condition-invariant signalthat, for motor cortex, is known to be tightly linked to the timingof movement onset5.

DiscussionIn highly interconnected networks, neural computations arebelieved to be instantiated by population-level dynamics59,69,70.We found that SMA and motor cortex differed in nearly everyaspect of their dynamics. SMA was less-well described by lineardynamics. To the degree that dynamical structure was present inSMA it was not dominated by rotations. In particular, 1.5–3 Hzrotational structure was absent in SMA, despite being prevalent inmotor cortex. These differences are notable because similardynamics might have been expected for at least three reasons.First, single-neuron responses in SMA and motor cortex appearbroadly similar during delayed-reach tasks, in both the presentand prior studies17,21,27,31. Indeed, prior studies specificallysought clear differences between SMA and motor cortex duringnon-sequential reaches, but did not find them. Second, anappealing idea is that different areas apply a canonical compu-tation to area-specific information34,35. Finally, it has remainedcontroversial whether rotational dynamics in motor cortex reflecta specific computation, or are a generic property of complex,high-dimensional data64. The present results make clear thatclasses of dynamics can be specific to different areas.

SMA responses appeared disorganized in the dynamicaldimensions. This is expected if responses do not obey the range ofhypotheses embodied in the HDR cost function. Ideally, wewould have employed other cost functions that embodied otherhypotheses. However, neither past nor present results have yetyielded sufficiently concrete hypotheses regarding SMA to allowthis strategy. That said, aspects of the present findings—incombination with prior work—suggest broad hypotheses thatcould be further refined. First, SMA possessed signals that couldpotentially contribute slower, non-multiphasic, features of muscleactivity. Such a contribution is plausible given that SMA con-tributes to the corticospinal tract. Second, SMA activity reflectedthe behavioral constraints governing when movement should beinitiated. Notably, such signals were present even during move-ment, after initiation. These observations suggest that the com-putations in SMA respect the broader context in whichmovement is executed.

Both SMA and motor cortex shared a condition-invariant shiftin state. For both areas the condition-invariant shift was large,contributing almost as much variance as the first two PCs. Such ashift is not an inevitable consequence of surface-level responsefeatures5 (e.g., overall increases in firing rate). For example,muscles activity often showed an overall increase across condi-tions, yet the muscle population exhibited a weak condition-invariant shift. We interpret the condition-invariant shift asreflecting neural events related to triggering movement. Forexample, in motor cortex, the timing of the condition-invariantshift is highly predictive of trial-by-trial variability in movementonset5. Furthermore, the condition-invariant shift is predicted bynetwork models5, where it does indeed serve the function ofrecruiting movement-generating dynamics.

The present data reveal that the shift is not only invariant withreach direction, it is also invariant with context. This is consistentwith the tentative hypothesis that more “cognitive” computations(perhaps involving SMA) take context into account, but that thefinal signal that triggers movement onset no longer containscontext information. That said, our data do not speak to thecausal flow. The shift could arise in SMA and be immediatelycommunicated to motor cortex, or the reverse could be true. It isalso plausible that both areas inherit the shift from a common




input, or that the shift is generated by global dynamics in whichmany areas participate. These possibilities are difficult to resolvebased on present observations. Synaptic latencies and transmis-sion times occur on a timescale (a few milliseconds) an order ofmagnitude faster than the timescale on which shift unfolds (tensto hundreds of milliseconds), making it difficult to infer causalityfrom chronology.

Our principal goal in applying HDR was to compare SMA andmotor cortex. However, the use of HDR also clarifies the pre-valence of rotational dynamics within motor cortex. The dyna-mical dimensions sought by HDR could have captured any typeof linear dynamical structure: rotations, contractions, expansions,shear, or other forms of non-normal dynamics. In particular,expansions would be predicted if motor cortex signals encode aconsistent preferred direction (e.g., if responses encode handvelocity or position). Instead, the dynamical dimensions formotor cortex were dominated by rotational structure, consistentwith the emergence of rotational dynamics in network modelstrained to produce muscle activity42. In accord with this inter-pretation, dynamical dimensions for motor cortex containedfeatures that matched, and thus may be contributing to, multi-phasic aspects of muscle activity.

In summary, our results demonstrate that neural responses inSMA and motor cortex appear similar in many ways, including atthe single-neuron level and when analyzed via standard methods.At the population level, both areas share a condition-invariantsignal. Yet there are large differences that become apparent whenfocusing on dynamics. Furthermore, population activity in thetwo areas covaries differently with muscle activity and with taskconstraints. These findings argue that SMA and motor cortex areprocessing different types of information via different dynamics,presumably with different computational goals.

MethodsSubjects and task. Subjects were two adult male macaque monkeys (Macacamulatta) aged 10 and 14 years and weighing 11–13 kg. Daily fluid intake wasregulated to maintain motivation to perform the task. All procedures were inaccord with the US National Institutes of Health guidelines and were approved bythe Columbia University Institutional Animal Care and Use Committee. Subjectssat in a primate chair facing an LCD display and performed reaches with their rightarm while their left arm was comfortably restrained. Hand position was monitoredusing an infrared optical system (Polaris; Northern Digital) to track (~0.3 mmprecision) a reflective bead temporarily affixed to the third and fourth digits.

We employed a center-out reaching task with three “contexts”44. Briefly, eachtrial began when the monkey touched and held a central touch-point. After thetouch-point was held for 450–550 ms (randomized) a colored 10 mm diameter disc(the target) appeared in one of eight possible locations radially arranged around thetouch point. In each context, similar reaches were made but the cue to initiate thereach was different. Touch-point and target color indicated context: red for cue-initiated, blue for self-initiated, and yellow for quasi-automatic. Target distance was130 mm for cue and self-initiated contexts and began at 40 mm for the quasi-automatic context (final reach distance was similar in all contexts—see below).Trials for different contexts/directions were randomly interleaved.

In the cue-initiated context, after a variable delay period (0–1000 ms) the targetsuddenly grew in size (to 30 mm) and the central touch point simultaneouslydisappeared. These events served as the go-cue, instructing the monkey to make themovement. Reaches were successful if they were initiated within 500 ms of the gocue, had a duration <500 ms, and landed within an 18 mm radius window centeredon the target. Juice was delivered if the monkey held the target, with minimal handmotion, for 200 ms (this criterion was shared across all three contexts).

In the self-initiated context, the target slowly and steadily grew in size, startingupon its appearance and ending when the reach began. Growth continued to amaximum size of 30 mm, which was achieved 1200 ms after target appearance(most reaches occurred before this time). The reward for a correct reach grewexponentially starting at 1 drop and achieved a maximum of 8 drops after 1200 ms.Monkeys were free to move as soon as the target appeared, but in practice nearlyalways waited some time: essentially all reaches occurred in the range from 600 to1400 ms after target onset. In rare instances where no movement was detectedwithin 1500 ms after target onset, the trial was aborted and flagged as an error.

In the quasi-automatic context, the target moved radially away from the centraltouch-point at 25 cm/s. Target motion began after a 0–1000 ms randomized delayperiod beginning at target onset. Target motion ended if a reach succeeded inbringing the hand to the target mid-flight. If the target was not intercepted (e.g., if

reach initiation was too slow) then the target continued moving until off the screen.Target speed and initial location (40 mm from the touch-point) were titrated,during training, such that the target was typically intercepted ~130 mm from thetouch-point (the same location as the targets for the other two contexts). Forsuccessful interception, reaches had to land within an elliptical acceptance window(16 mm by 20mm radius, with the long axis aligned with target motion). If thetarget was successfully intercepted, it grew in size to 30 mm and reward wasdelivered after the hold period.

In the present study, we analyze only trials where the delay period, for cue-initiated and quasi-automatic contexts, was >400 ms. Trials with shorter delayswere included to encourage immediate and robust preparation, and because for thepurposes of another study44 we were interested in zero-delay trials in the quasi-automatic context. However, in the present study we wished to examinemovement-period dynamics following the establishment of preparatory activity, inkeeping with refs. 5–7,42.

Neural and muscle recordings. After subjects became proficient in the task, weperformed sterile surgery to implant a head restraint. At the same time, weimplanted a recording chamber centered over the arm area of motor cortex of theleft hemisphere, including primary motor cortex (M1) and the dorsal premotorcortex (PMd). After recordings from M1/PMd were complete, the chamber wasremoved and a new chamber was implanted over the left-hemisphere SMA.Chamber positioning was guided by structural magnetic resonance images takenshortly before implantation. We used intracortical microstimulation to confirmthat our recordings were from the forelimb region of motor cortex (biphasic pulses,cathodal leading, 250 µS pulse width delivered at 333 Hz for a total duration of 50ms). Microstimulation of motor cortex typically evoked contractions of theshoulder and upper-arm muscles, at currents from 5 to 60 μA depending on thelocation and cortical layer. Microstimulation of SMA (total duration of 200 ms)sometimes caused full-arm movements reminiscent of a reach or an intentionalarm movement at currents of ~20–100 µA. Other times, microstimulation of SMAup to 150 µA did not elicit any movement. As expected, thresholds were oftenhigher in SMA. We thus used longer trains of microstimulation (and generallyhigher currents) in SMA simply because this was more effective in evokingmovement, and we wished to verify that we were in arm-related SMA.

We recorded single-neuron responses using traditional tungsten electrodes(FHC) or one or more silicon linear-array electrodes (V-probes; Plexon) loweredinto cortex using a motorized microdrive. For tungsten-electrode recordings, spikeswere sorted online using a window discriminator (Blackrock Microsystems). Forlinear-array recordings, spikes were sorted offline (Plexon Offline Sorter). Werecorded all well-isolated task-responsive neurons; no attempt was made to screenfor neuronal selectivity for reach direction or any other response property. Spikeswere smoothed with a Gaussian kernel with standard deviation of 20 ms andaveraged across trials to produce peri-stimulus time histograms.

We recorded electromyogram (EMG) activity using intramuscular electrodesfrom the following muscles: lower and upper aspects of the trapezius, medial,lateral and anterior aspects of the deltoid, medial and outer aspects of the biceps,brachialis, pectoralis, and latismus dorsi. The triceps were found to be minimallyactive (consistent with our prior observations using similar tasks) and were notrecorded further. EMG signals were bandpass filtered (10–500 Hz), digitized at 1kHz, rectified, smoothed with a Gaussian kernel with standard deviation of 20 ms,and averaged across trials to produce peri-stimulus time histograms.

Population vector. We first defined a preferred direction for each neuron byregressing movement-epoch firing rates (averaged over a 500 ms window centeredon movement onset) versus horizontal and vertical target location. This was donefor all 24 conditions (eight directions and three contexts). The population vectorfor a given condition was the sum of these direction vectors, each weighted by theaverage firing rate of the corresponding neuron.

Preprocessing. Prior to dimensionality reduction, each neuron’s response wassoft-normalized so that neurons with high firing rates had approximately unityfiring-rate range (normalization factor= firing rate range+ 5 spikes/s). This stepfollows our standard approach (e.g., refs. 5,6,10), and ensures that the identifieddimensions attempt to capture the response of all neurons, rather than a handful ofhigh-firing-rate neurons. This is particularly important because many dimen-sionality reduction techniques (including PCA, HDR, and jPCA) focus on cap-turing variance. Without soft-normalization, a neuron with a firing rate of75 spikes/s would contribute 25 times more variance than a neuron with a firingrate of 15 spikes/s.

Hypothesis-guided dimensionality reduction. Dimensionality reduction beganby formatting neural (or muscle) responses as a matrix, A, where each columncontains the responses of one neuron, concatenated across all times and conditions.A is thus of size CT×N where C is the number of conditions, T is the number oftime points and N is the number of neurons. We also consider A, where eachelement describes the derivative of the firing rate for the corresponding condition,time and neuron. We seek a CT×K matrix, X, where each column describes one ofK latent variables (K was set to six for all analyses). X is found via projection:





X=AWT, where W is a K×N orthogonal matrix. Each latent variable is thus aweighted sum of individual-neuron responses, with the weights defining Kdimensions in neural space. Those weights were found by optimizing a costfunction:W ¼ argmin

Wf Wð Þ, with the constraint thatW is orthonormal. We used a

tripartite cost function: f(W)=frec(W)+finvar(W)+fdyn(W), with each term corre-sponding to one aspect of the guiding hypothesis.

The first term, frec(W), is identical to the cost function used by PCA, andencourages the dimensions in W to capture variance in A, such that A can bereconstructed reasonably accurately from X. Because W is an orthogonal matrix,Arec=XW=AWTW is the optimal linear reconstruction of A (in terms ofminimizing mean squared error) from X. Thus,

frec Wð Þ ¼ A� Areck k2fro¼ A� AWTW�� 2

fro. The squared Frobenius norm, �k k2fro ,

is the sum of the squares of the individual elements. frec(W) is small if A can bereasonably well reconstructed from X. PCA can be thought of as a special case ofHDR, where only the term frec(W) is used. This corresponds to the 0th orderhypothesis that the largest signals (those that most dominate the responses of singleneurons) are important.

The second and third terms relate to the hypothesis that there exists condition-invariant structure in some dimensions and dynamical structure in other,orthogonal dimensions. We consider W to be partitioned into two parts: W=[Winvar;Wdyn]. This results in a partitioned

X ¼ Xinvar;Xdyn

h i¼ A WT

invar;WTdyn

h i¼ AWT. The second and third terms of the

cost function relate to Xinvar and Xdyn, respectively.The second term of the cost function, finvar(W), is small if Xinvar ¼ AWT

invar isinvariant across conditions. We setfinvar Wð Þ ¼ trace WinvarCacrossW

Tinvar

� �=trace WinvarCindW

Tinvar

� �. Where, Cind is the

covariance matrix describing the aspects of A that are condition-independent, andCacross is the covariance matrix describing the aspects of A that vary acrossconditions. Cacross is constructed by computing the covariance after removing, foreach neuron, the cross-condition mean: the average firing rate at each time acrossall conditions. Cind is the covariance of the cross-condition mean itself. finvar(W), isthus small if, for the latent variables in Xinvar, the cross-condition mean variesstrongly with time but there is little variance among conditions around that mean.

The third term of the cost function, fdyn(W), attempts to identify a Wdyn wherethe resulting Xdyn ¼ AWT

dyn and its temporal derivative, _Xdyn ¼ _AWTdyn are linearly

related, such that _Xdyn � XdynD for some D. Assume that D is chosen to provide

the best fit, which can be accomplished by setting D ¼ Xydyn

_Xdyn, where † indicates

the pseudo-inverse. Then the variance accounted for by the fit is XdynD��

��2

fro. To

find a W that maximizes this variance, we set

fdyn Wð Þ ¼ �XdynD��

��2

fro¼ � XdynX

ydyn

_Xdyn

��2

fro¼ � ðAWT

dynÞðAWTdynÞyð _AWT

dynÞ��

��2

fro

. fdyn(W) is thus small if the dimensions in WTdyn capture structure whose temporal

evolution (for all conditions) is well described by a linear dynamical system.Minimizing f(W) thus produces projections that balance capturing maximal

data variance, finding a set of dimensions where trajectories are similar acrossconditions, and finding another set of dimensions where trajectories are fit by alinear dynamical system.

Iterative optimization is required to find the minimum-cost projection matrixW. Full details of the optimization technique can be found in ref. 67. In brief, wewish to take gradient steps in the objective function f(W) while respecting theconstraint that W is an orthogonal matrix (W belongs to the Stiefel manifold). Todo so, we first project the gradient ∇f ðWÞ onto the tangent space of the constraintmanifold, step in that direction, and then project the result back onto the constraintmanifold. Although not guaranteed to reach the global optima (since the constraintmanifold is nonconvex), this optimization is provably convergent to a localoptimum. In practice we found the lack of global guarantee was not a majorconcern: for the datasets we analyzed, re-running optimization multiple times withdifferent initializations resulted in final W that spanned very similar spaces. For thedatasets analyzed here, optimization converged relatively rapidly (~1 s on a 2017-era Apple Macbook Pro running Matlab 2016b).

Finding the best-fit purely rotational linear dynamics. For some analyses, wewished to ask how well trajectories in the dynamical dimensions could be describedby purely rotational linear dynamics, if rotational dynamics were fit directly. To do

so, we found D�skew ¼ argmin

D

_Xdyn � XdynD��

��fro, subject to the constraint that D=

−DT (i.e., that D is skew-symmetric). �k kfro indicates the Frobenius norm, which inthis case is simply the root-mean-squared error of the fit6. This is equivalent toperforming regression, but with the constraint that the fit is provided by a purelyrotational system. The eigenvalues of D�

skew are imaginary and were used tocompute the rotational frequencies in Fig. 6c, d.

Bootstrap tests for statistical significance. Dimensionality reduction yieldedlatent variables with different properties for different datasets. For example,although HDR always sought dynamical dimensions where _Xdyn � XdynD, thegoodness of this fit (the R2) varied between SMA, motor cortex, and muscle

populations. To ask whether R2 differed between areas, one might be tempted tosimply regress _Xdyn against Xdyn and compare the traditional confidence limits onR2. However, this approach will overstate significance because, for both Xdyn and_Xdyn, rows are not independent (e.g., nearby times tend to have similar states andsimilar derivatives). We therefore sought alternative approaches.

First, we employed a bootstrap in which we redrew, with replacement, 24 newconditions from the original 24. Each column of Xdyn and _Xdyn was modified toinclude data from the 24 redrawn conditions. We then recomputed the R2. Thisprocess was repeated 1000 times to provide the sampling distribution. The p-valuefor a given comparison was the number of draws where the effect was not observed:e.g., if the R2 for motor cortex was greater than the R2 for SMA for both the originaldata and for 995/1000 bootstrap draws, then p=0.005.

The bootstrap described above accounts for the possibility that the R2 for onedataset might appear larger than that for another dataset due to “random”differences in individual-condition trajectories. This is reasonable, as the quality ofthe dynamical fit is in large part determined by whether different conditions obeythe same dynamical flow-field. However, this approach does not address a differentconcern: perhaps the group of neurons recorded from one area simply happened tohave more dynamical structure. This concern could be addressed by redrawingneurons, but that would not address the larger concern that different patches ofcortex might be more or less “dynamical”, e.g., perhaps motor cortex recordingssimply happened to encounter a more dynamical group of neurons than did SMArecordings. To address this potential concern, we employed a very conservativebootstrap. This approach treated the four dynamical dimensions recorded fromone area (e.g., motor cortex) and the four dynamical dimensions recorded fromanother area (e.g., SMA) as constituting eight dimensions in one largerundifferentiated “area”. We then drew four random dimensions from this eight-dimensional space, and computed the R2. This was done twice, and we computedthe difference in R2. We then collected a distribution of such differences across1000 repetitions. This procedure asks how often one would observe a largedifference in R2 if there were truly no difference other than a random bias in whichdimensions were sampled. The p-value was the number of such differences thatwere as large or larger than the empirical difference, e.g., if randomdifferences were smaller than the empirical difference for 995/1000 repetitions,then p=0.005.

jPCA- and PCA-based approaches to assessing dynamics. We applied the jPCAalgorithm as described in ref. 6. This involved three steps. First, for each neuron thecross-condition mean was removed such that the average firing rate (across con-ditions) was zero. Second, PCA was applied and the projection onto the top six PCswas retained. Third, we found the best-fit rotational linear dynamics (see above)that described the evolution of activity in those dimensions. Subsequent analysisfocused on the features of these linear rotational dynamics, including fit quality androtation frequency. The removal of the cross-condition mean (the first step) wasimportant to ensure that PCA found dimensions where activity co-varies acrossconditions. Without this step, the first two dimensions are typically close to con-dition invariant. Notably, HDR did not employ this initial step because condition-invariant and condition-specific structure were isolated via a different (and gen-erally preferable) method: by projection onto orthogonal dimensions. The PCA-based approach was nearly identical to the jPCA-based approach, but in the thirdstep we fit with an unconstrained linear dynamical system. For both approaches,we applied a bootstrap procedure. The neural population was redrawn 100 timeswith replacement, and the analysis was repeated each time. This allowed us to putconfidence intervals on measures such as the dynamical fit and the rotation fre-quencies, and to plot the eigenvalue spectra for multiple bootstrap repetitions toillustrate when values did or did not cluster.

Code availability. Code related to the dimensionality reduction approach of ref. 67

is provided at: https://github.com/cunni/ldr. Optimization code specific tothe present study is available upon request to the corresponding author.The jPCA code package is available from the Churchland laboratory website:http://churchlandlab.neuroscience.columbia.edu

Data availability. All datafiles used to produce the figures and analyses in thismanuscript are available, in matlab format, by direct request made to the corre-sponding author.

Received: 5 December 2017 Accepted: 11 June 2018

References1. Evarts, E. V. Relation of pyramidal tract activity to force exerted during

voluntary movement. J. Neurophysiol. 31, 14–27 (1968).2. Ashe, J. & Georgopoulos, A. P. Movement parameters and neural activity

in motor cortex and area 5. Cereb. Cortex 4, 590–600 (1994).



https://github.com/cunni/ldr

http://churchlandlab.neuroscience.columbia.edu


3. Sergio, L. E. & Kalaska, J. F. Changes in the temporal pattern of primary motorcortex activity in a directional isometric force versus limb movement task.J. Neurophysiol. 80, 1577–1583 (1998).

4. Reimer, J. & Hatsopoulos, N. G. The problem of parametric neural coding inthe motor system. Adv. Exp. Med. Biol. 629, 243–259 (2009).

5. Kaufman, M. T. et al. The largest response component in the motor cortexreflects movement timing but not movement type. eNeuro 3, (2016).

6. Churchland, M. M. et al. Neural population dynamics during reaching.Nature 487, 51–56 (2012).

7. Churchland, M. M., Cunningham, J. P., Kaufman, M. T., Ryu, S. I. & Shenoy,K. V. Cortical preparatory activity: representation of movement or first cog ina dynamical machine? Neuron 68, 387–400 (2010).

8. Kaufman, M. T., Churchland, M. M., Ryu, S. I. & Shenoy, K. V. Corticalactivity in the null space: permitting preparation without movement. Nat.Neurosci. 17, 440–448 (2014).

9. Fetz, E. E. Are movement parameters recognizably coded in the activity ofsingle neurons? Behav. Brain Sci. 15, 679–690 (1992).

10. Seely, J. S. et al. Tensor analysis reveals distinct population structure thatparallels the different computational roles of areas M1 and V1. PLoS Comput.Biol. 12, e1005164 (2016).

11. Shenoy, K. V., Sahani, M. & Churchland, M. M. Cortical control of armmovements: a dynamical systems perspective. Annu. Rev. Neurosci. 36,337–359 (2013).

12. Kao, J. C. et al. Single-trial dynamics of motor cortex and their applications tobrain-machine interfaces. Nat. Commun. 6, 7759 (2015).

13. Pandarinath, C. et al. Neural population dynamics in human motor cortexduring movements in people with ALS. eLife 4, e07436 (2015).

14. Petreska, B. et al. in Advances in Neural Information Processing Systems, Vol.24 (eds Shawe-Taylor, J. et al.) 756–764 (Curran Associates, Inc., Red Hook,New York, 2011).

15. Picard, N. & Strick, P. L. Motor areas of the medial wall: a review of theirlocation and functional activation. Cereb. Cortex 6, 342–353 (1996).

16. Nachev, P., Kennard, C. & Husain, M. Functional role of the supplementaryand pre-supplementary motor areas. Nat. Rev. Neurosci. 9, 856–869 (2008).

17. Tanji, J. & Mushiake, H. Comparison of neuronal activity in thesupplementary motor area and primary motor cortex. Brain. Res. Cogn. Brain.Res. 3, 143–150 (1996).

18. Shima, K. & Tanji, J. Neuronal activity in the supplementary andpresupplementary motor areas for temporal organization of multiplemovements. J. Neurophysiol. 84, 2148–2160 (2000).

19. Eccles, J. C. The initiation of voluntary movements by the supplementarymotor area. Arch. Psychiatr. Nervenkr. 231, 423–441 (1982).

20. Passingham, R. E. The Frontal Lobes and Voluntary Action (Oxford UniversityPress, Oxford, 1993).

21. Chen, D. F., Hyland, B., Maier, V., Palmeri, A. & Wiesendanger, M.Comparison of neural activity in the supplementary motor area and in theprimary motor cortex in monkeys. Somatosens. Mot. Res. 8, 27–44 (1991).

22. Mushiake, H., Inase, M. & Tanji, J. Selective coding of motor sequence in thesupplementary motor area of the monkey cerebral cortex. Exp. Brain Res. 82,208–210 (1990).

23. Thaler, D., Chen, Y. C., Nixon, P. D., Stern, C. E. & Passingham, R. E. Thefunctions of the medial premotor cortex. I. Simple learned movements. Exp.Brain Res. 102, 445–460 (1995).

24. Fried, I. et al. Functional organization of human supplementary motor cortexstudied by electrical stimulation. J. Neurosci. 11, 3656–3666 (1991).

25. Goldberg, G. Supplementary motor area structure and function - review andhypotheses. Behav. Brain Sci. 8, 567–588 (1985).

26. Brinkman, C. & Porter, R. Supplementary motor area in the monkey: activityof neurons during performance of a learned motor task. J. Neurophysiol. 42,681–709 (1979).

27. Okano, K. & Tanji, J. Neuronal activities in the primate motor fields of theagranular frontal cortex preceding visually triggered and self-pacedmovement. Exp. Brain Res. 66, 155–166 (1987).

28. Thaler, D. E., Rolls, E. T. & Passingham, R. E. Neuronal activity of thesupplementary motor area (SMA) during internally and externally triggeredwrist movements. Neurosci. Lett. 93, 264–269 (1988).

29. Picard, N. & Strick, P. L. Activation of the supplementary motor area (SMA)during performance of visually guided movements. Cereb. Cortex 13, 977–986(2003).

30. Alexander, G. E. & Crutcher, M. D. Preparation for movement: neuralrepresentations of intended direction in three motor areas of the monkey. J.Neurophysiol. 64, 133–150 (1990).

31. Kurata, K. & Wise, S. P. Premotor and supplementary motor cortex in rhesusmonkeys: neuronal activity during externally- and internally-instructed motortasks. Exp. Brain Res. 72, 237–248 (1988).

32. Romo, R. & Schultz, W. Neuronal activity preceding self-initiated or externallytimed arm movements in area 6 of monkey cortex. Exp. Brain Res. 67,656–662 (1987).

33. Munoz, D. P. & Schall, J. D. in The Oculomotor System: New Approaches forStudying Sensorimotor Integration. (eds Hall, W. C. & Moschovakis, A. K.)52–82 (CRC Press, Boca Raton, FL, USA, 2003).

34. Harris, K. D. & Shepherd, G. M. The neocortical circuit: themes andvariations. Nat. Neurosci. 18, 170–181 (2015).

35. Douglas, R. J., Martin, K. A. C. & Whitteridge, D. A canonical microcircuitfor neocortex. Neural Comput. 1, 480–488 (1989).

36. Dum, R. P. & Strick, P. L. Motor areas in the frontal lobe of the primate.Physiol. Behav. 77, 677–682 (2002).

37. Boudrias, M. H., Belhaj-Saif, A., Park, M. C. & Cheney, P. D. Contrastingproperties of motor output from the supplementary motor area and primarymotor cortex in rhesus macaques. Cereb. Cortex 16, 632–638 (2006).

38. Hummelsheim, H., Bianchetti, M., Wiesendanger, M. & Wiesendanger, R.Sensory inputs to the agranular motor fields: a comparison betweenprecentral, supplementary-motor and premotor areas in the monkey.Exp. Brain Res. 69, 289–298 (1988).

39. Scangos, K. W., Aronberg, R. & Stuphorn, V. Performance monitoring bypresupplementary and supplementary motor area during an arm movementcountermanding task. J. Neurophysiol. 109, 1928–1939 (2013).

40. Perez, M. A., Tanaka, S., Wise, S. P., Willingham, D. T. & Cohen, L. G. Time-specific contribution of the supplementary motor area to intermanual transferof procedural knowledge. J. Neurosci. 28, 9664–9669 (2008).

41. Churchland, M. M. & Cunningham, J. P. A dynamical basis set for generatingreaches. Cold Spring Harb. Symp. Quant. Biol. 79, 67–80 (2014).

42. Sussillo, D., Churchland, M. M., Kaufman, M. T. & Shenoy, K. V. A neuralnetwork that finds a naturalistic solution for the production of muscle activity.Nat. Neurosci. 18, 1025–1033 (2015).

43. Hall, T. M., de Carvalho, F. & Jackson, A. A common structure underlies low-frequency cortical dynamics in movement, sleep, and sedation. Neuron 83,1185–1199 (2014).

44. Lara, A. H., Elsayed, G. F., Cunningham, J. & Churchland, M. M.Conservation of preparatory neural events regardless of how movement isinitiated. bioRrxiv, https://doi.org/10.1101/189035 (2017).

45. Churchland, M. M. & Shenoy, K. V. Temporal complexity and heterogeneityof single-neuron activity in premotor and motor cortex. J. Neurophysiol. 97,4235–4257 (2007).

46. Hatsopoulos, N. G., Xu, Q. & Amit, Y. Encoding of movement fragments inthe motor cortex. J. Neurosci. 27, 5105–5114 (2007).

47. Wise, S. P., Weinrich, M. & Mauritz, K. H. Movement-related activity in thepremotor cortex of rhesus macaques. Prog. Brain. Res. 64, 117–131 (1986).

48. Kaufman, M. T. et al. Roles of monkey premotor neuron classes in movementpreparation and execution. J. Neurophysiol. 104, 799–810 (2010).

49. Rickert, J., Riehle, A., Aertsen, A., Rotter, S. & Nawrot, M. P. Dynamicencoding of movement direction in motor cortical neurons. J. Neurosci. 29,13870–13882 (2009).

50. Crammond, D. J. & Kalaska, J. F. Prior information in motor and premotorcortex: activity during the delay period and effect on pre-movement activity.J. Neurophysiol. 84, 986–1005 (2000).

51. Georgopoulos, A. P., Schwartz, A. B. & Kettner, R. E. Neuronal populationcoding of movement direction. Science 233, 1416–1419 (1986).

52. Gallego, J. A., Perich, M. G., Miller, L. E. & Solla, S. A. Neural manifolds forthe control of movement. Neuron 94, 978–984 (2017).

53. Stevenson, I. H. et al. Functional connectivity and tuning curves inpopulations of simultaneously recorded neurons. PLoS Comput. Biol. 8,e1002775 (2012).

54. Goris, R. L., Movshon, J. A. & Simoncelli, E. P. Partitioning neuronalvariability. Nat. Neurosci. 17, 858–865 (2014).

55. Sadtler, P. T. et al. Neural constraints on learning. Nature 512, 423–426(2014).

56. Gao, Y., Archer, E., Paninski, L. & Cunningham, J. P. 30th Conference onNeural Information Processing Systems (NIPS) (Barcelona, Spain, 2016).

57. Macke, J. H. et al. Neural Information Processing Systems (NIPS) (Lake Tahoe,CA, USA, 2016).

58. Machens, C. K., Romo, R. & Brody, C. D. Functional, but not anatomical,separation of “what” and “when” in prefrontal cortex. J. Neurosci. 30, 350–360(2010).

59. Mante, V., Sussillo, D., Shenoy, K. V. & Newsome, W. T. Context-dependentcomputation by recurrent dynamics in prefrontal cortex. Nature 503, 78–84(2013).

60. Stopfer, M., Jayaraman, V. & Laurent, G. Intensity versus identity coding in anolfactory system. Neuron 39, 991–1004 (2003).

61. Briggman, K. L. & Kristan, W. B. Multifunctional pattern-generating circuits.Annu. Rev. Neurosci. 31, 271–294 (2008).

62. Michaels, J. A., Dann, B., Intveld, R. W. & Scherberger, H. Predicting reactiontime from the neural state space of the premotor and parietal graspingnetwork. J. Neurosci. 35, 11415–11432 (2015).

63. Cunningham, J. P. & Yu, B. M. Dimensionality reduction for large-scaleneural recordings. Nat. Neurosci. 17, 1500–1509 (2014).



https://doi.org/10.1101/189035



64. Elsayed, G. F. & Cunningham, J. P. Structure in neural population recordings:an expected byproduct of simpler phenomena? Nat. Neurosci. 20, 1310–1318(2017).

65. Michaels, J. A., Dann, B. & Scherberger, H. Neural populationdynamics during reaching are better explained by a dynamical systemthan representational tuning. PLoS Comput. Biol. 12, e1005175(2016).

66. Kobak, D. et al. Demixed principal component analysis of neural populationdata. eLife 5, e10989 (2016).

67. Cunningham, J. P. & Ghahramani, Z. Linear dimensionality reduction: survey,insights, and generalizations. J. Mach. Learn. Res., 16, 2859-2900 (2015).

68. Churchland, M. M., Yu, B. M., Sahani, M. & Shenoy, K. V. Techniques forextracting single-trial activity patterns from large-scale neural recordings.Curr. Opin. Neurobiol. 17, 609–618 (2007).

69. Barak, O., Sussillo, D., Romo, R., Tsodyks, M. & Abbott, L. F. From fixedpoints to chaos: three models of delayed discrimination. Prog. Neurobiol. 103,214–222 (2013).

70. Sussillo, D. & Barak, O. Opening the black box: low-dimensional dynamics inhigh-dimensional recurrent neural networks. Neural Comput. 25, 626–649(2013).

AcknowledgementsWe thank Yanina Pavlova and Sean Perkins for technical support. This work was sup-ported by the Sloan Foundation, the Simons Foundation (SCGB#325233 andSCGB#542957), the Grossman Center for the Statistics of Mind, the McKnight Foun-dation, NINDS (1DP2NS083037), NIH CRCNS R01NS100066, NINDS 1U19NS104649,P30 EY019007, a Klingenstein-Simons Fellowship, the Searle Scholars Program, and anNIH Postdoctoral Fellowship (F32 NS092350).

Author contributionsAll authors collaborated in defining the general approach and key analyses. A.H.L.and M.M.C. designed the experiments. A.H.L. ran the experiments and recorded the

data. J.P.C. implemented the optimization that allowed the central HDR analysis. A.H.L.and M.M.C. performed the analyses. All authors participated in interpreting results andwriting the manuscript.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-05146-z.

Competing interests: The authors declare no competing interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,

adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is not included in thearticle’s Creative Commons license and your intended use is not permitted by statutoryregulation or exceeds the permitted use, you will need to obtain permission directly fromthe copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

© The Author(s) 2018



https://doi.org/10.1038/s41467-018-05146-z

https://doi.org/10.1038/s41467-018-05146-z

http://npg.nature.com/reprintsandpermissions/

http://npg.nature.com/reprintsandpermissions/

http://creativecommons.org/licenses/by/4.0/

http://creativecommons.org/licenses/by/4.0/


OPEN Different population dynamics in the supplementary ...stat.columbia.edu/~cunningham/pdf/LaraNCOMMS2018.pdf · SMA and motor cortex both shared a large signal previously shown

Documents